Theorem grlimprop2 47889

Description: Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)

Hypotheses

Ref	Expression
grlimprop.v	⊢ 𝑉 = (Vtx‘𝐺)
grlimprop.w	⊢ 𝑊 = (Vtx‘𝐻)
grlimprop2.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
grlimprop2.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
grlimprop2.i	⊢ 𝐼 = (iEdg‘𝐺)
grlimprop2.j	⊢ 𝐽 = (iEdg‘𝐻)
grlimprop2.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
grlimprop2.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

Assertion

Ref	Expression
grlimprop2	⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Distinct variable groups:   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉   𝑓,𝐹,𝑔,𝑣   𝑓,𝐺,𝑔,𝑖,𝑥   𝑓,𝐻,𝑔,𝑖,𝑥   𝑥,𝐼   𝑥,𝐽   𝑖,𝐾   𝑖,𝐿   𝑓,𝑀,𝑔,𝑖,𝑥   𝑓,𝑁,𝑔,𝑖,𝑥   𝑣,𝑖

Allowed substitution hints:   𝐹(𝑥,𝑖)   𝐼(𝑣,𝑓,𝑔,𝑖)   𝐽(𝑣,𝑓,𝑔,𝑖)   𝐾(𝑥,𝑣,𝑓,𝑔)   𝐿(𝑥,𝑣,𝑓,𝑔)   𝑀(𝑣)   𝑁(𝑣)   𝑉(𝑥,𝑓,𝑔,𝑖)   𝑊(𝑥,𝑣,𝑓,𝑔,𝑖)

Proof of Theorem grlimprop2

Step	Hyp	Ref	Expression
1		grlimdmrel 47883	. . . . 5 ⊢ Rel dom GraphLocIso
2	1	ovrcl 7472	. . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
3		id 22	. . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
4		df-3an 1088	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) ↔ ((𝐺 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
5	2, 3, 4	sylanbrc 583	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
6		grlimprop.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
7		grlimprop.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐻)
8		grlimprop2.n	. . . 4 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣)
9		grlimprop2.m	. . . 4 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))
10		grlimprop2.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
11		grlimprop2.j	. . . 4 ⊢ 𝐽 = (iEdg‘𝐻)
12		grlimprop2.k	. . . 4 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
13		grlimprop2.l	. . . 4 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
14	6, 7, 8, 9, 10, 11, 12, 13	isgrlim2 47886	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
15	5, 14	syl 17	. 2 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
16	15	ibi 267	1 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ⊆ wss 3963  dom cdm 5689   “ cima 5692  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  Vtxcvtx 29028  iEdgciedg 29029   ClNeighbVtx cclnbgr 47743   GraphLocIso cgrlim 47879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-vtx 29030  df-iedg 29031  df-clnbgr 47744  df-isubgr 47785  df-grim 47802  df-gric 47805  df-grlim 47881

This theorem is referenced by: (None)